Why doesn't the set $\{ -2, +2 \}$ in $E(X) = (y − \text{sign}\{\overline{W} \cdot \overline{X} \}) \in \{ −2, +2 \}$ include $0$?
It is important to note that the exact statement is the eqation given below can never be 0 for misclassified points in $ S^+$
$$ E(X) = (y - \text{sign}\{\overline{W} \cdot \overline{X}\}) $$
And $S+$ is defined as the set of all misclassified training points $X \in S$ that satisfy the condition $y(\overline{W} \cdot \overline{X})&lt;0 $ which means that $y$ and $ (\overline{W} \cdot \overline{X}) $ will have different signs i.e if $y &lt; 0$ then $(\overline{W} \cdot \overline{X}) &gt; 0$ and vice versa.
Finally the condition on $S^+$ constraints the the $ E(X)$ to these values when $ y &lt; 0$ and $(\overline{W} \cdot \overline{X}) &gt; 0$
$$ E(X) = (-1 - \text{sign}(+ve)) = -2 $$
or when $ y &gt; 0$ and $(\overline{W} \cdot \overline{X}) &lt; 0$
$$ E(X) = (1 - \text{sign}(-ve)) = 2 $$